Proof for tabular integration?

In summary, tabular integration is a shortcut method for integration by parts. However, there is not a specific proof for tabular integration, as it is simply a shorthand method. By using placeholder functions for u and v, it can be seen that the process is the same as using tabular integration. This method is simply a way to save time by remembering the tabular way.
  • #1
MadViolinist
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Today we learned tabular integration as a shortcut method for integration by parts. Is there a proof that legitimizes tabular integration out there, or some general formula? Because there has to be some sort of logic for the process of creating the diagonal lines, or assigning pluses and minuses. Thanks in advance.
 
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  • #2
There isn't really a proof for tabular integration, it's just shorthand for integration by parts. If you did a integration by parts using placeholder functions for u and v (meaning the actual functions could be whatever you want) you'll see that it expands to the same thing that tabular integration does, it just saves time doing by remembering the tabular way.
 

1. What is tabular integration?

Tabular integration is a method used to numerically approximate the value of a definite integral. It involves dividing the interval of integration into smaller subintervals and using the trapezoidal rule to estimate the area under the curve within each subinterval.

2. How does tabular integration work?

To perform tabular integration, the interval of integration is first divided into equally spaced subintervals. The trapezoidal rule is then used to calculate the area under the curve within each subinterval. Finally, the calculated areas are added together to approximate the value of the definite integral.

3. What are the benefits of using tabular integration?

Tabular integration is a relatively simple and straightforward method for approximating definite integrals. It is also more accurate than other numerical integration methods, such as the midpoint rule or Simpson's rule, when the function being integrated is not smooth.

4. Are there any limitations to tabular integration?

Tabular integration is limited by the number of subintervals that can be used to approximate the integral. If the interval of integration is too large or the function being integrated is highly oscillatory, the accuracy of the approximation may be reduced. In these cases, other numerical integration methods may be more suitable.

5. How can I check the accuracy of my tabular integration results?

One way to check the accuracy of tabular integration results is to compare them to the exact value of the definite integral, if it is known. Another method is to use a smaller interval of integration and compare the results to the original approximation. Additionally, mathematical software or calculators can be used to perform the integration and compare the results to those obtained through tabular integration.

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